Remoteness and distance eigenvalues of a graph

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• 作者：Huiqiu Lin^a ; ^{huiqiulin@126.com" class="auth_mail" title="E-mail the corresponding author} ; Kinkar Ch. Das^b ; ^{kinkardas2003@googlemail.com" class="auth_mail" title="E-mail the corresponding author} ; Baoyindureng Wu^c ; ^{wubaoyin@hotmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Remoteness ; Distance matrix ; Distance eigenvalues
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：31 December 2016
• 年：2016
• 卷：215
• 期：Complete
• 页码：218-224
• 全文大小：369 K

文摘

Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303389&_mathId=si7.gif&_user=111111111&_pii=S0166218X16303389&_rdoc=1&_issn=0166218X&md5=0e4723cea6ce5286c885b70e51a3ba7a" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$ be a connected graph of order class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303389&_mathId=si8.gif&_user=111111111&_pii=S0166218X16303389&_rdoc=1&_issn=0166218X&md5=311663609dc8945755827b8f75785fa3" title="Click to view the MathML source">nclass="mathContainer hidden">class="mathCode">$n$ with diameter class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303389&_mathId=si4.gif&_user=111111111&_pii=S0166218X16303389&_rdoc=1&_issn=0166218X&md5=c6aa73f8e717c2d21c4cd4fc1b650fa5" title="Click to view the MathML source">dclass="mathContainer hidden">class="mathCode">$d$. Remoteness class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303389&_mathId=si10.gif&_user=111111111&_pii=S0166218X16303389&_rdoc=1&_issn=0166218X&md5=7f5efe289cf4264f63e04d9110b5d64a" title="Click to view the MathML source">ρclass="mathContainer hidden">class="mathCode">$\rho$ of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303389&_mathId=si7.gif&_user=111111111&_pii=S0166218X16303389&_rdoc=1&_issn=0166218X&md5=0e4723cea6ce5286c885b70e51a3ba7a" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$ is the maximum average distance from a vertex to all others and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303389&_mathId=si12.gif&_user=111111111&_pii=S0166218X16303389&_rdoc=1&_issn=0166218X&md5=b26950d15938b7bb8ba0c437ba8863a0" title="Click to view the MathML source">∂₁≥⋯≥∂_nclass="mathContainer hidden">class="mathCode">${\partial }_1\ge \cdots \ge {\partial }_n$ are the distance eigenvalues of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303389&_mathId=si7.gif&_user=111111111&_pii=S0166218X16303389&_rdoc=1&_issn=0166218X&md5=0e4723cea6ce5286c885b70e51a3ba7a" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$. Aouchiche and Hansen (0000), Aouchiche and Hansen conjectured that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303389&_mathId=si14.gif&_user=111111111&_pii=S0166218X16303389&_rdoc=1&_issn=0166218X&md5=0f21067e0b8e755eb6a958c5e9605b28" title="Click to view the MathML source">ρ+∂₃>0class="mathContainer hidden">class="mathCode">$\rho +{\partial }_3>0$ when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303389&_mathId=si15.gif&_user=111111111&_pii=S0166218X16303389&_rdoc=1&_issn=0166218X&md5=fca7c18df0e2c6b0e628a9398102ddc5" title="Click to view the MathML source">d≥3class="mathContainer hidden">class="mathCode">$d\ge 3$ and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303389&_mathId=si16.gif&_user=111111111&_pii=S0166218X16303389&_rdoc=1&_issn=0166218X&md5=7f221e512880917e256eac002e770347">class="imgLazyJSB inlineImage" height="21" width="94" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X16303389-si16.gif">class="mathContainer hidden">class="mathCode">$\rho +{\partial }_{\lfloor \frac{7d}{8}\rfloor }>0$. In this paper, we confirm these two conjectures. Furthermore, we give lower bounds on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303389&_mathId=si17.gif&_user=111111111&_pii=S0166218X16303389&_rdoc=1&_issn=0166218X&md5=c33b029178bb116c43b82e0657b30ab1" title="Click to view the MathML source">∂_n+ρclass="mathContainer hidden">class="mathCode">${\partial }_n+\rho$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303389&_mathId=si18.gif&_user=111111111&_pii=S0166218X16303389&_rdoc=1&_issn=0166218X&md5=2055f7da7b5a152259df1442fc1e5389" title="Click to view the MathML source">∂₁−ρclass="mathContainer hidden">class="mathCode">${\partial }_1-\rho$ when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303389&_mathId=si19.gif&_user=111111111&_pii=S0166218X16303389&_rdoc=1&_issn=0166218X&md5=3efb5d3ed957229c693486688c4ab1e3" title="Click to view the MathML source">G≇K_nclass="mathContainer hidden">class="mathCode">$G\ncong K_n$ and the extremal graphs are characterized.